The measurements by Zagarola & Smits (1998) of mean velocity profiles in fully developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the uncertainties due to velocity gradient corrections. A new static pressure correction (McKeon & Smits 2002) is used in analysing all data and leads to significant differences from the Zagarola & Smits conclusions. The results confirm the presence of a power-law region near the wall and, for Reynolds numbers greater than $230\,{\times}\,10^3$ ($R^+\,{>}\,5\,{\times}\,10^3$), a logarithmic region further out, but the limits of these regions and some of the constants differ from those reported by Zagarola & Smits. In particular, the log law is found for $600\,{<}\, y^+\,{<}\,0.12R^+$ (instead of $600\,{<}\,y^+\,{<}\,0.07R^+$), and the von Kármán constant $\kappa$, the additive constant $B$ for the log law using inner flow scaling, and the additive constant $B^*$ for the log law using outer scaling are found to be $0.421 \pm 0.002$, $5.60 \pm 0.08 $ and $1.20 \pm 0.10$, respectively, with 95% confidence level (compared with $0.436 \pm 0.002$, $6.15 \pm 0.08$, and $1.51 \pm 0.03$ found by Zagarola & Smits). The data also confirm that the pipe flow data for Re$_D\,{\le}\,13.6\,{\times}\,10^6$ (as a minimum) are not affected by surface roughness.